Nonlinear finite elements/Bubnov Galerkin method

(Bubnov)-Galerkin Method for Problem 2Edit

The Bubnov-Galerkin method is the most widely used weighted average method. This method is the basis of most finite element methods.

The finite-dimensional Galerkin form of the problem statement of our second order ODE is :

 

Since the basis functions ( ) are known and linearly independent, the approximate solution   is completely determined once the constants ( ) are known.

The Galerkin method provides a great way of constructing solutions. But the question is: how do we choose   so that these functions are not only linearly independent but arbitrary? Since the solution is expressed as a sum of these functions, the accuracy of our result depends strongly on the choice of  .

Let the trial solution take the form,

 

According to the Bubnov-Galerkin approach, the weighting function also takes a similar form

 

Plug these values into the weak form to get

 

or

 

or

 

Taking the sums and constants outside the integrals and rearranging, we get

 

Since the  s are arbitrary, the quantity inside the square brackets must be zero. That is

 

Let us define

 

Then we get a set of simultaneous linear equations

 

In matrix form,